# How to construct explicit homotopies for maps of simplicial sets?

Let's say I have two simplicial sets $K_\bullet$ and $L_\bullet$ and two maps of simplicial sets $f,g:K_\bullet\to L_\bullet$. Are there standard techniques to construct homotopies between two such maps (if they are indeed homotopic, of course)? Or at least, is there a good way to describe $\Delta[1]\times K_\bullet$?

Remark: I know how to do it in the case where $K_\bullet$ is a standard simplex, as I did it in my bachelor's thesis (page 45). However simplicial sets can only be described as colimits (namely, coequalizers) of standard simplices, and as far as I know limits and colimits do not commute...

In the specific case I'm interested in, $L_\bullet = K_\bullet$, $g = 1_{K_\bullet}$, and I have an explicit description for $f$. (Yeah, I want to show that a second simplicial set is homotopy equivalent to $K_\bullet$.)

If $L_\bullet$ is a Kan complex (and it is the case for me), then we can consider right homotopies, and that will be the same as left homotopies. Therefore, we must find $$H:K_\bullet\longrightarrow \operatorname{Hom}(\Delta[1],L_\bullet)$$ satisfying adequate boundary conditions (see e.g. Goerrs-Jardine, Simplicial homotopy theory, page 77), where $\operatorname{Hom}$ is the function complex for simplicial sets (Goerrs-Jardine, page 21).