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.

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    $\begingroup$ (Aside: somewhere in the paper it's presumably important that $V$ is symmetric monoidal closed with coequalizers -- or at least that $v\otimes (-)$ preserves coequalizers for each $v \in V$.) I haven't gone into the details, but 2.9(ii) is a strange statement. Rarely is such a diagram relevant. I would think that Wolff needs it to show that $VCat$ is monadic over $VGph$ using the Beck monadicity theorem, in which case he probably only needs it for a split coequalizer, which has an extra equation. $\endgroup$
    – Tim Campion
    Feb 5, 2022 at 21:01
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    $\begingroup$ To be more explicit, if the point is to verify the hypotheses of the Beck monadicity theorem, it's probably sufficient to assume the additional equation $GHF = GHG$, making $G,H,F$ into what's called a "contractible pair" on the above-linked nlab page. The resulting data can be thought of as a 1-truncated simplicial object with extra degeneracies. $\endgroup$
    – Tim Campion
    Feb 5, 2022 at 21:12
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    $\begingroup$ Ah, sorry, "split coequalizer" or "contractible pair" in the terminology of the nlab page I linked to is different from what Wolff calls a "split coequalizer" in 2.9(i) (which I would rather call a "reflexive coequalizer). Any coequalizer arising from an adjunction will almost surely be contractible in the sense of the nlab page I linked to. It's funny that Wolff should structure his argument that way -- the fact that $VCat$ has coequalizers follows anyway from monadicity (since $VGph$ is cocomplete), and nowhere in proving monadicity does one need to understand general coequalizers. $\endgroup$
    – Tim Campion
    Feb 5, 2022 at 21:31
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    $\begingroup$ As alluded to, there's a completely general formula for any category of algebras of a monad. Unfortunately, for $VCat$, coequalizers in general are nasty, so to get anywhere useful one generally wants to take into account more information about whatever particular coequalizer one is interested in. Also, coproducts + pushouts + filtered colimits also give all colimits -- I find the pushout formula in $VCat$ slightly easier to grok than the coequalizer one. $\endgroup$
    – Tim Campion
    Feb 5, 2022 at 21:39
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    $\begingroup$ Filtered colimits are created by the forgetful functor to $VGph$. This follows from monadicity + the fact that the free $V$-category monad preserves filtered colimits. That is, a filtered colimit in $VCat$ is computed by taking the filtered colimit of the underlying graphs, and then defining composition in the obvious way. $\endgroup$
    – Tim Campion
    Feb 7, 2022 at 13:08


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