Is isomorphism defined between large categories? By definition, an isomorphism between two objects in a category is a morphism so and so.. We know that $\mathbf {Cat}$ is the categories of small categories so that morphisms between objects are functors. Now that we have no category of large categories (or do we have?!), how are we going to define an isomorphism between two large categories? 
Or like many other cases (and mainly because of foundational reasons) we again ignore such entities and prefer to talk about small/locally small categories?
Thanks.
 A: Everything depends on foundations you work in. 
The first way to think about it is that you can define the notion of isomorphism of categories (or any other types of objects) without mentioning a category containing this isomorphism. For example, in the following way:
Definition. Let $\mathcal{A}$ and $\mathcal{B}$ be categories, $T\colon\mathcal{A}\to\mathcal{B}$ be a functor. Then we say that $T$ is an isomorphism iff there exists such functor $S\colon\mathcal{B}\to\mathcal{A}$, that $S\circ T=I_{\mathcal{A}}$ and $T\circ S=I_{\mathcal{B}}$. 
The second way is to take appropriate foundations. There is actually no need to deal with the category of all small categories. You may fix a Grothendieck universe $\mathcal{U}$ and define the category $\mathbf{Cat}_{\mathcal{U}}$ of all $\mathcal{U}$-small categories. Then $\mathbf{Cat}_{\mathcal{U}}$ is not a $\mathcal{U}$-small category, but it is a $\mathcal{U'}$-small category for some universe $\mathcal{U'}$, such that $\mathcal{U}\in\mathcal{U'}$. In such case, for instance, the isomorphism of duality 
$$
\text{op}_{\mathcal{U}}\colon\mathbf{Cat}_{\mathcal{U}}\to\mathbf{Cat}_{\mathcal{U}}
$$
such that $\text{op}(\mathcal{A})=\mathcal{A}^{op}$, is an isomorphism in $\mathbf{Cat}_{\mathcal{U'}}$.
