Definition of the cotensor in sSet

I am currently trying to understand the meaning of the cotensor in sSet. Suppose I have X,Y objects in sSet, the cotensor is denoted by $X^Y$. I found that for sSet the cotensor is a map of the form $$Set^{op}\times sSet\to sSet$$ However, I would like a more concrete definition and intuition of the cotensor.

Looking at different constructions (which were not in sSet), I thought that $X^Y$ might be something of the form $X^Y(n)=Hom_{-}(Y,X(n))$. But I do not see how the Hom could be taken over sSet since $X(n)$ is now just a set. So then I thought maybe something like all possible inclusions of the set $X(n)$ into the different n-simplices, but I don't yet see how that would work out.

Does anyone know how this is defined and am I on the right track?

A good reference would be appreciated.

As you say, the cotensor is a functor $$ \mathrm{Set}^{\text{op}} \times \mathrm{sSet} \to \mathrm{sSet} $$ It sends a set $X$ and a simplicial set $S_{\bullet}$ to a simplicial set $S_{\bullet}^{X}$ defined by the following universal property: $$ \mathrm{sSet}(T_{\bullet}, S_{\bullet}^{X}) \cong \mathrm{Set}(X, \mathrm{sSet}(T_{\bullet}, S_{\bullet})) $$ But for any sets $X$ and $Y$ we have $$ \mathrm{Set}(X, Y) \cong \prod_{X} Y $$ and so in this case we can also describe the cotensor by the universal property $$ \mathrm{sSet}(T_{\bullet}, S_{\bullet}^{X}) \cong \prod_{X} \mathrm{sSet}(T_{\bullet}, S_{\bullet}) \cong \mathrm{sSet}\left(T_{\bullet}, \prod_{X} S_{\bullet}\right) $$ or, by Yoneda, $$ S_{\bullet}^{X} = \prod_{X} S_{\bullet} $$ This description didn't use any properties of $\mathrm{sSet}$, so it works just as well for any other category cotensored over set. As for references, I learned about cotensors from Riehl's Categorical Homotopy Theory, $\S3.7$, which you might also find helpful.