I study algebraic geometry / number theory and from time to time I stumble upon 2-categorical (co)limits. I have two main examples in mind:

Example 1) In étale cohomology, the (triangulated) derived category of $\mathbb{Q}_\ell$-sheaves is defined as a 2-colimit of the derived categories of sheaves with coefficients in finite extensions of $\mathbb{Q}_\ell$.

Example 2) The fact that $\textsf{QCoh}\to \textsf{Sch}$ is a stack should mean that, given a scheme $X$ and a covering $\{U_i\to X\}$, the category $\textsf{QCoh}(X)$ is a 2-limit of the $\textsf{QCoh}(U_i)$.

While the second example is somewhat straightforward, given that we may describe the limit as a category of descent data, the first one is awkward.

For sure, I shouldn't need to understand a lot of 2-category theory to make sense of these (and all related) examples. There are a lot of intricacies in the 2-categorical world... For example, should we consider (co)limits in the (2,0)-category of categories or on the (2,1)-category of categories? Should we consider lax 2-functors or strict 2-functors? What changes with those choices? (Bear in mind that I know very little of all of this.)

All in all, how should an algebraic geometer approach these kinds of statements? Also, is there a quick reference for all of this?

I study algebraic geometry / number theory and from time to time I stumble upon 2-categorical (co)limits. I have two main examples in mind:

Example 1) In étale cohomology, the (triangulated) derived category of $\overline{\mathbb{Q}_\ell}$-sheaves is defined as a 2-colimit of the derived categories of sheaves with coefficients in finite extensions of $\mathbb{Q}_\ell$.

Example 2) The fact that $\textsf{QCoh}\to \textsf{Sch}$ is a stack should mean that, given a scheme $X$ and a covering $\{U_i\to X\}$, the category $\textsf{QCoh}(X)$ is a 2-limit of the $\textsf{QCoh}(U_i)$.

While the second example is somewhat straightforward, given that we may describe the limit as a category of descent data, the first one is awkward.

For sure, I shouldn't need to understand a lot of 2-category theory to make sense of these (and all related) examples. There are a lot of intricacies in the 2-categorical world... For example, should we consider (co)limits in the (2,0)-category of categories or on the (2,1)-category of categories? Should we consider lax 2-functors or strict 2-functors? What changes with those choices? (Bear in mind that I know very little of all of this.)

All in all, how should an algebraic geometer approach these kinds of statements? Also, is there a quick reference for all of this?