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In this paper we show that if $\mathscr{C}$ is a category and if$F\colon\mathscr{C} \to \mathfrak{Cat}$ is a pseudofunctor such that for eachobject $X$ of $\mathscr{C}$ the category $F(X)$ is a tangent category and foreach morphism $f$ of $\mathscr{C}$ the functor $F(f)$ is part of a strongtangent morphism $(F(f),\!{}_{f}{\alpha})$ and that furthermore the naturaltransformations $\!{}_{f}{\alpha}$ vary pseudonaturally in$\mathscr{C}^{\operatorname{op}}$, then there is a tangent structure on thepseudolimit $\mathbf{PC}(F)$ which is induced by the tangent structures on thecategories $F(X)$ together with how they vary through the functors $F(f)$. Weuse this observation to show that the forgetful $2$-functor$\operatorname{Forget}:\mathfrak{Tan} \to \mathfrak{Cat}$ creates and preservespseudolimits indexed by $1$-categories. As an application, this allows us todescribe how equivariant descent interacts with the tangent structures on thecategory of smooth (real) manifolds and on various categories of (algebraic)varieties over a field.

Pseudolimits for Tangent Categories and Equivariant Tangents for Varieties and Smooth Manifolds