Let us consider homomorphisms between partial algebras as defined in
There, an isomorphism from $A$ to $B$ is a
bijective homomorphism from $A$ to $B$
such that its inverse is a homomorphism, too.
I can see that for partial algebras the second condition is necessary. Is the second condition also necessary if $A$ is a total algebra, or can this condition be dropped? I.e., is condition 2 implied by condition 1 when $A$ is total?