The name Monshu Gate Keeper in Chinese / Japanese...

In modern Japanese, 門主 is the title of a head priest of a temple or monastery.

This can also refer to the spiritual leader of the sect and/or direct descendant of its founder.
In the past, this could refer to the founder of a Buddhist sect.

The literal meaning of 門主 is gatekeeper or keeper of the gate.

In the Chinese Buddhist dictionary, this entry comes up as the controller of a gate or sect. However, this term is not commonly used in Chinese.

大智慧門 is the title for “The Great Wisdom Door,” or “The Gate to Great Wisdom.”

This refers to a portal to all wisdom and knowledge in Buddhism.

門 is the Chinese, Japanese Kanji, and old Korean Hanja for gate, door, gateway, doorway, opening, entrance, valve, or switch.

Most commonly, this is simply a gate or door.

門 is also a Chinese surname that romanizes as Men.

In Japanese, this can be surnames romanizing as Yuki, Mon, To, or Kado.

In the Buddhist context, beyond meaning a door or gate, this can be a counter for a Buddhist sect, teaching, or school (we might say “slice” of bread, this would be a “gate” of teaching). In this way, it kind of refers to one of several doors that lead to salvation or nirvana.

This is the Dharma Gate, The Gate to Enlightenment, or Dharmaparyāya.

The doctrines, or wisdom of Buddha is regarded as the door to enlightenment.

安樂の法門 represents that the Dharma Gate or the Door to Enlightenment leads to peace and happiness.

The Dharma Gate is the method of attaining or actualizing peace and happiness in Zen and other Buddhist traditions.

大道無門 is a Buddhist proverb that means “The Great Way has no entrance,” “The Great Way is gateless,” or “The Great Path lacks a gate.”

This can be translated in many other ways.

This concept was authored within a long sacred text by 無門慧開 (known as Wúmén Huìkāi in Chinese or Mumon Ekai in Japanese). He was a Chinese Chan Master (in Japanese, a Zen Master) who lived between 1183 and 1260 AD. His most famous work was a 48-koan collection titled “The Gateless Barrier” or “The Gateless Gate” (無門關 Wú Mén Guān in Chinese, or 無門関 Mu Mon Kan in Japanese). This calligraphy title is a notable line from this collection.

I like this reference to the source of this proverb: The Gateless Gate 無門關

無門關 is the ancient title for “The Gateless Barrier.” This has both direct meaning and is the title of a 13th-century collection of koans, compiled by a Chinese Chan Master known as Wumen in China (known in Japan as Zen Master Mumon).

While this is the original title, you may see this written as 無門関 in Japan, where the last character, 關, was simplified to 関 after 1945.

無門関 is the specifically Japanese title for “The Gateless Barrier.” This has both direct meaning and is the title of a 13th-century collection of koans, compiled by Zen Master Mumon (actually a Chinese Chan Master known as Wumen in China).

The original title is 無門關, but the last Kanji was simplified to 関 in modern Japan.

黑旗門 is often romanized as “Hek Ki Boen” in Cantonese, this is the title for a certain branch school of Wing Chun or Eng Chun martial arts.

開門 means “to open a door,” “opening gate” or figuratively, “to open for business.”

守護者 is a Chinese, Japanese Kanji, and old Korean Hanja title for a protector, watcher, or keeper.

The character breakdown:
守護 = to guard; to protect; protection; safeguard.
者 = person; someone of that nature; someone doing that work, the one [who, which].

From this breakdown, you can translate this as “The one who protects,” “Someone who safeguards,” or “guardian.”

悶 is a character that can mean melancholy, sorrow, or depression.

The visual meaning depicts a heart shut behind a door or gate. In some contexts outside of melancholy, this can mean sealed, airtight, or shut indoors.

This sad or sorrowful character is kind of a strange selection for a calligraphy wall scroll.

During the Tang Dynasty, a man named Jia Dao (born in the year 779), a well-studied scholar and poet, went to the capital to take the imperial examination.

One day as he rides a donkey through the city streets, a poem begins to form in his mind. A portion of the poem comes into his head like this:

“The bird sits on the tree branch near a pond,
A monk approaches and knocks at the gate...”

At the same time, he wondered if the word “push” would be better than “knock” in his poem.

As he rides down the street, he imagines the monk pushing or knocking. Soon he finds himself making motions of pushing and shaking a fist in a knocking motion as he debates which word to use. He is quite a sight as he makes his way down the street on his donkey with hands and fists flying about as the internal debate continues.

As he amuses people along the street, he becomes completely lost in his thoughts and does not see the mayor's procession coming in the opposite direction. Jia Bao is blocking the way for the procession to continue down the road, and the mayor's guards immediately decide to remove Jia Bao by force. Jia Bao, not realizing that he was in the way, apologizes, explains his poetic dilemma and awaits his punishment for blocking the mayor's way.

The mayor, Han Yu, a scholar and author of prose himself, finds himself intrigued by Jia Dao's poem and problem. Han Yu gets off his horse and addresses Jia Bao, stating, “I think knock is better.” The relieved Jia Bao raises his head and is invited by the mayor to join the procession, and are seen riding off together down the street, exchanging their ideas and love of poetry.

In modern Chinese, this 反復推敲 idiom is used when someone is trying to decide which word to use in their writing or when struggling to decide between two things when neither seems to have a downside.