$V$-cat and $V$-graph: coequalizers in the category of enriched functors

This question is regarding the 1974 JPAA paper $V$-cat and $V$-graph by Harvey Wolff.

To be precise, I don’t understand a certain step in the proof of Corollary 2.9, which (the corollary) is crucial for the main theorem of the paper.

Let $V$ be a closed symmetric monoidal category with coequalizers, $A$ an $V$-enriched category and $U$ the forgetful functor from $V$-enriched categories to $V$-enriched graphs. A pair $F,G\colon D\to U(A)$ is called a pre-$V$-congruence in $A$ if $Ob(D) = Ob(A)$ ($D$ is a $V$-graph and $A$ is a $V$-category). Given a pre-$V$-congruence $(F,G)$, there is an associated $V$-graph $E: Ob(E) = Ob(A)$ and $E(A,B)$ is the coequalizer of $F_{A,B},G_{A,B}\colon D(A,B)\to A(A,B)$ in $V$. It comes with an associated morphism $L$ of $V$-graphs where $L_{A,B}$ is the coequalizer map. This graph is a coequalizer of $F$ and $G$ in the category $V$-graphs. A pre-$V$-congruence in a $V$-congruence if $E$ is a $V$-category is a way that $L$ is a $V$-enriched functor.

Now Corollary 2.9.(ii) says that if $F_1,F_2\colon A\to B$ are $V$-functors such that $(U(F_1),U(F_2))$ is a pre-$V$-congruence, for which there exists a $V$-functor $H\colon B\to A$ satisfying $F_1H = 1$, then it is a $V$-congruence.

Now for the step I don’t understand: the author goes from $M_B\circ (1\otimes (F_2H))$ to $F_2H\circ M_B$ where $M_B$ is the composition of morphisms in $B$. I’ve got a hunch that either I’m missing something obvious and making a fool of myself posting this or there is a mistake, but an enriched functor should satisfy $M_B\circ((F_2H)\otimes (F_2H)) = (F_2H)\circ M_B$, and not what is written there.

If I indeed am missing something trivial, please, be gentle: I’m not great at enriched category theory, and my only interest of these questions is understanding why the category of small dg-categories is cocomplete.

Edit. If the proof is indeed incorrect, can it still be salvaged? Alternatively, even if it’s not the question per se, I would still be satisfied with the excplicit construction in the case $V = Ch(k)$, in the case of dg-categories and dg-functors.

Answer

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Source : Link , Question Author : Jxt921 , Answer Author : Community

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