Here we study the algebraic properties of the proof theory of intuitionist first-order logic in a categorical setting. Our work is based on the confluence of ideas and techniques from proof theory, category theory, and combinatory logic, and this book is addressed to specialists in all three areas.Proof theorists will find that categories give rise to a non-trivial semantics for proof theory in which the concept of the equivalence of proofs can be investigated from a mathematical point of view. Categorists, on the other hand, will find that proof theory provides a suitable syntax in which commutative diagrams can be characterized and classified effectively. Workers in combinatory logic, finally, may derive new insights from the study of algebraic invariance properties of their techniques established in the course of our presentation.