I don't know anything about the subject itself, but according to math. review 93a:03062 there's a paper by Barry Jay refuting one of Szabo's claims. The paper is `Coherence in category theory and the Church-Rosser property', Notre Dame J Formal Logic 33 (1992), no 1, 140-143. Tom Todd Wilson wrote:
In the book
M. E. Szabo, Algebra of Proofs, Studies in Logic and the Foundations of Mathematics, Vol. 88, North-Holland, 1978.
[...]
The book itself contains a wealth of technical detail that includes many dozens of claims whose proofs are not worked out in detail. I'm writing to ask whether anyone has any knowledge about the degree to which this work was refereed and/or whether the results have been verified independently. I'm appealing especially to those who work in categorical logic or those interested in automatic proof verification in category theory, both of which groups, it would seem, should be interested in Szabo's work.