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To transfer a tensored and cotensored simplicially enriched structure from a category $\mathcal{C}$ to $(\mathcal{C}\downarrow Z)$, we define $(X\to Z)\otimes K$ by the composite $(X\otimes K \to X \otimes \{0\} \to Z)$ and $(X\to Z)^K$ by a pullback of $X^K\to Z^K$ and $Z^{\{0\}} \to Z^K$. I just need to know in what textbook it is written down for a bibliographical reference (I don't want to reinvent the wheel). It is also not in Overcategories and undercategories of cofibrantly generated model categories.

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    $\begingroup$ My own preference -- define the simplicial enrichment on $C \downarrow Z$ by $Hom(X \xrightarrow f Z, Y \xrightarrow g Z) = Hom(X,Y) \times_{Hom(X,Z)} \{f\}$ and then derive the tensoring and cotensoring formulas via computations from this definition. Even better, the simplicial hom formula that I just wrote down can itself be derived as a computation if you define the simplicially-enriched category $C \downarrow Z$ by the appropriate 2-categorical universal property in the 2-category of simplicially-enriched categories (which is well-known to be 2-categorically complete and cocomplete). $\endgroup$
    – Tim Campion
    Commented Mar 8, 2022 at 16:04

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