(Co)tensoring of enriched slice categories

Question

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.

Answer 1

Your guess for the copower (née tensor) is correct. You can check it by giving it the right universal property with respect to the enriched homs as described in the question you linked. Similarly, for the power (née cotensor) you can check that $g^K$ for $g:B\to X$ can be defined by the pullback of the induced map $B^K \to X^K$ along $X \cong X^{1_V} \to X^K$.

However, I don't know offhand how to generalize away from the semicartesian case. Possibly the right point of view in that case is that the slice $C/X$ is not enriched over $V$ itself but over $V/I$; I think that it can be given powers and copowers for that enrichment.