FWIW, I like John’s description above (and probably object much less than baseline to humorously confrontational language in research contexts :). I agree that for most math contexts, using the standard definitions with morphism sets and composition mappings is easier to prove things with, but I think the intuition described here is great and often in better agreement with how mathematicians intuit about category-theoretic constructions than the explicit formalism.
FWIW, I like John’s description above (and probably object much less than baseline to humorously confrontational language in research contexts :). I agree that for most math contexts, using the standard definitions with morphism sets and composition mappings is easier to prove things with, but I think the intuition described here is great and often in better agreement with how mathematicians intuit about category-theoretic constructions than the explicit formalism.