I think a good answer to this comes from category theory, linear logic, diagrams and the geometry of tensor calculus (Joyal-street). We often talk about category theory through the use of diagrams which are planar graphs (objects as nodes morphisms as arrows). Written down, these diagrams can be seen as fishing nets, kind of embedded in a plane, so we don't care about how one line crosses over another. These diagrams can be rewritten in different ways that respect the topology of the network. These deformations are exactly what Joyal and Street were talking about in the geometry of tensor calculus. We know from further work that, (and excuse my poor explanation ) the geometry of tensor calculus is a model of linear logic. This would mean that the axioms of a symmetric monoidal category can support the axioms of linear logic...(please excuse the poor understanding, I have come to these thoughts with little help). The long and short is that, if we talk about category theory in terms of diagrams, then we are most likely thinking in terms of linear logic. This would be in contrast to a model of the theory of categories in Set. In that case, we have a set of objects and set of morphisms, the axioms of (some kind of ) Set theory and further the axioms of the category we want to talk about. This would be thinking in a different kind of logic.