Category theory is also, in some sense, a continuation of the work of Emmy Noether (one of Mac Lane's teachers) in formalizing abstract processes; Noether realized that in order to understand a type of mathematical structure, one needs to understand the processes preserving that structure.
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More recent efforts to introduce undergraduates to categories as a foundation for mathematics include William Lawvere and Rosebrugh (2003) and Lawvere and Stephen Schanuel (1997) and Mirroslav Yotov (2012).
Three model-theoretic directions for universal logic have been explored to some depth: abstract model theory axiomatized by Jon Barwise, a topological / categorical approach based on sketches (sometimes called categorical model theory), and yet another categorical approach based on Goguen and Burstall's notion of institution.
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A. Joyal, Ross Street, An introduction to Tannaka duality and quantum groups, Category theory (Como, 1990), 413—492, Lecture Notes in Math.
Natural transformation in category theory, a branch of abstract mathematics