  Associated to a presentable $\infty$-category $\mathcal{C}$ and an object $X
\in \mathcal{C}$ is the tangent $\infty$-category $\mathcal{T}_X\mathcal{C}$,
consisting of parameterized spectrum objects over $X$. This gives rise to a
cohomology theory, called Quillen cohomology, whose category of coefficients is
$\mathcal{T}_X\mathcal{C}$. When $\mathcal{C}$ consists of algebras over a nice
$\infty$-operad in a stable $\infty$-category, $\mathcal{T}_X\mathcal{C}$ is
equivalent to the $\infty$-category of operadic modules, by work of
Basterra--Mandell, Schwede and Lurie. In this paper we develop the
model-categorical counterpart of this identification and extend it to the case
of algebras over an enriched operad, taking values in a model category which is
not necessarily stable. This extended comparison can be used, for example, to
identify the cotangent complex of enriched categories, an application we take
up in a subsequent paper.
