Lemma 2.4 in chapter 2 of Goerss Jardine Simplicial homotopy theory

Question

I am having trouble with lemma 2.4 of the chapter of simplicial categories. I have a feeling there are mistakes in the proof or maybe typos. He wants to prove the 2.1.2 property of the definition of a simplicial category, that is Hom$_{C}(A,-)\colon C\to sSets$ is a right adjoint of $A\otimes-\colon sSets\to C$. The problem is I cannot understand why he takes an coequalizer of a simplicial set (and how this is even done). Moreover, in his notation, he has the notation hom$_{C}(A,B)$, where by assumption we have a map hom$_{C}(K,-)\colon C\to C$ and $K\in sSets$. However, since he has a tensor with $A$ in the previous line with the standard simplices, then it should be $A\in C$, so there is a problem with the definition of the functor hom$_{C}(A,B)$.

Answer 1

They (there are two authors) are taking a coequalizer because, by assumption (2), the tensor product functor commutes with all colimits, and hence will commute with the coequalizer. The functor $\mathbf{hom}_{\mathcal{C}}(K, \cdot)$ is a right adjoint by assumption, so it commutes with limits.