Or an alternate title: using evil for the greater good.

In category theory, the principle of equivalence says that statements about *things* should be invariant under the appropriate notion of *thing*-equivalence. For example, if we have a group $G$ we shouldn't ask a question like "does the underlying set of $G$ contain $x$" because it's possible to have $x \in G$ and $x \notin H$ while $G \cong H$. One formalization of this principle is the univalence axiom.

I think the principle of equivalence is a reasonable guiding principle, but care must be taken when determining "the appropriate notion of *thing*-equivalence". The classic example is that categories themselves naturally assemble into a $2$-category, not just an ordinary $1$-category, and so the suitable notion of equivalence (equivalence of categories) is weaker than naïve version (isomorphism of categories). As a consequence, the definition "A group is a groupoid with a single object" (Joke 1.1 in Algebra: Chapter 0 by Paolo Aluffi) violates the principle of equivalence, because the property of "having a single object" is not invariant under equivalence. We can remedy this to the definition "A group is a pointed connected groupoid" (although perhaps even the notion of a pointed groupoid violates the principle of equivalence, so we really want to say it's an object of the under $2$-category $T/\mathsf{Grpd}$ whose underlying groupoid is connected, where $T$ is "the" $2$-terminal groupoid). Contemplating this does lead to some interesting ideas, e.g. if we view groups as groupoids in a naïve way they're more like torsors.

But I like groups! And I like doing algebra. Working with groupoids as strict, algebraic objects isn't inherently "evil". Ronald Brown's book Topology and Groupoids discusses cofibrations of groupoids, strict pushouts of groupoids, and groupoids defined by generators and relations, and I think it's genuinely interesting stuff. You certainly need to be careful when working with categories as strict objects, but I find it easier to think about strict things than weak things. I like that the framework of model categories lets me present weak things by strict things and I like working with strict $2$-categories over weak ones. Surely any category theorist would agree that the Grothendieck construction is of fundamental importance, and the usual statement of it in terms of Grothendieck fibrations violates the principle of equivalence (because Grothendieck fibrations involve equality of objects). It's no coincidence that the idea of a Grothendieck fibration was (I believe) historically prior to the idea of a Street fibration, or that strict $n$-categories are so much easier to define than weak ones. It's also frequently simpler to think about strict monoidal categories and strict monoidal functors than the maximally weak ones. We even have Lack's "Theorem", saying "naturally occurring bicategories tend to be equivalent to naturally occurring strict $2$-categories".

Although it's not always possible to get everything we want from strict categories (e.g. there is no strict $3$-groupoid equivalent as a weak $3$-category to $\Pi_{\leq 3} \mathbb{S}^2$) it can still be useful when studying categories to think about them strictly. See also the $(\mathrm{bo}, \mathrm{ff})-$factorization system, the canonical model structure on $\mathsf{Cat}$, the nerve construction, and the Thomason model structure on $\mathsf{Cat}$. What other examples are there *within category theory* where it's useful to treat categories themselves as strict objects? This question was motivated by looking at the paper "Amalgamations of Categories" by MacDonald and Scull (because I had a strict pullback square of categories and I wanted to know if it was also a strict pushout square).

Edit: One more use case I forgot to mention is the bar construction associated to a comonad. If $W$ is a comonad on $\mathsf{C}$ then we can view it as a comonoid object in the *strict monoidal* category $(\operatorname{Fun}(\mathsf{C}, \mathsf{C}), \circ, \operatorname{Id})$. This is classified by a unique strict monoidal functor $F : \Delta^{\mathrm{op}} \to \operatorname{Fun}(\mathsf{C}, \mathsf{C})$, where $\Delta^{\mathrm{op}}$ is the augmented simplex category under ordinal sum. We can view $F$ instead as a functor from $\mathsf{C}$ to the category of augmented simplicial objects of $\mathsf{C}$, and this functor takes an object to its simplicial bar resolution with respect to $W$. We could still basically make this work without strictness, since (I believe) if $\mathsf{M}$ is a not-necessarily strict monoidal category then a monoid object of $\mathsf{M}$ is classified by an *essentially* unique strong monoidal functor $\Delta \to \mathsf{M}$. But I find it easier to think about a unique strict monoidal functor than an essentially unique strong monoidal one.

specificobject you have to state things in language that would make sense if that object is replaced by anything equivalent to it. I would say it's only about dependence on variable objects. $\endgroup$