I was contemplating asking this question on Philsophy.SE but felt it was better directed here as there are a dearth of category theorists there.
According to the wikipedia entry on Categorical Logic:
Lawvere's writings, sometimes couched in a philosophical jargon, isolated some of the basic concepts as adjoint functors (which he explained as 'objective' in a Hegelian sense, not without some justification).
What was the explanation or justification.
The idea is that the process of describing a collection of mathematical objects by constructing a theory, and the process of finding all mathematical objects satisfying a mathematical theory, are adjoint operations in a precise sense. The back and forth dialectic between the concrete objects on the one hand and abstract theories on the other then translates to applying these adjoint functors, and composing them, perhaps with further adjoints.
In "any serious field of study", Lawvere takes 'objective logic' to mean the construction of concepts necessary for that study, and 'subjective logic' to mean making inferences between statements in a theory, subjective in the sense that any inference is relative to its assumptions.
All this is described for example in Lawvere's article Tools for the advancement of Objective Logic: Closed Categories and Toposes (1994) in the conference proceedings volume
The logical foundations of cognition, Vancouver Studies in Cognitive Science, edited by John Macnamara and Gonzalo E. Reyes