In an answer to this question: Enriched slice categories, a description of the enrichment of the slice category in an enriched category is given. I'm interested in going a bit further. If we assume that the original enriched category, say $\mathcal{C}$, is $\mathcal{V}$-enriched but also $\mathcal{V}$-tensored and cotensored, and that $\mathcal{V}$ is semi-Cartesian (i.e. the monoidal unit is terminal) how can we describe the tensoring and cotensoring of a slice $\mathcal{C}_{/X}$ for $X\in\mathcal{C}$? It seems to me that the tensoring should probably be some kind of weighted colimit, which I believe would mean we just compute it on the "underlying object." My immediate guess about what this would look like is the following: given an object $f:A\to X\in\mathcal{C}_{/X}$, the tensoring with $K\in\mathcal{V}$ is $$f\otimes K:A\otimes K\overset{A\otimes\varepsilon_K}\longrightarrow A\otimes 1_{\mathcal{V}}\cong A\overset{f}\to X$$

But first of all, I think this is probably wrong, and second of all, I'm wondering if there is an explicit description for the *cotensoring* as well, such that we have the usual isomorphisms

$$\mathit{Map}_{\mathcal{C}_{/X}}(f\otimes K, g)\cong \mathit{Hom}_\mathcal{V}(K,Map_{\mathcal{C}_{/X}}(f,g))\cong \mathit{Map}_{\mathcal{C}_{/X}}(f, g^K)$$

I happen to be only interested in the case that $\mathcal{V}=sSet$, but perhaps there's a more general answer. I'm not even sure we need the condition that $\mathcal{V}$ is semi-Cartesian honestly, since it seems like the statement about tensors being colimits, and the forgetful functor preserving colimits, wouldn't require it.