What does it mean to "identify" two objects in algebra, as distinct from having an isomorphism or automorphism between them?
I was led to think about this while reading Stewart's Galois Theory, where there are often sentences such as "A field extension is a monomorphism $\iota \colon K \twoheadrightarrow L$, where $K$, $L$ are fields. Usually we identify $K$ with its image $\iota(K)$, and in this case $K$ becomes a subfield of $L$."