"Identity-free" definition of an isomorphism in a semigroupoid / semicategory I am looking for a way to define "Isomorphism" in a semigroupoid (or semicategory), that is a "category", which does not necessarily have identities.
To be more specific I am looking for a way to make the following theorem work in semigroupoids:

Initial objects (Terminal objects) are unique up to isomorphism.

So, actually I am looking for a notion of objects "being isomorphic" in semigroupoids. However, I am also interested in how said isomorphisms may look like. 

I am still new to category theory and I am just asking this out of curiosity. I (briefly) read about "Equivalence of categories", which is kind of a relaxed "version" of categories being isomorphic, but as far as I know this definition only makes sense for objects in $\mathsf{Cat}$.

I really don't know, where my head was. It's simpler than it seemed:
Let $f$ be an isomorphism, if there exists a morphism $g$, such that $g\circ f$ is a one-sided identity.
Then we have:

Initial objects (Terminal objects) are unique up to isomorphism.

Proof:
Observe, that there is unique endomorphism $f$ on an initial object $0$. Then for all objects $C$ and morphisms $g : 0 \rightarrow C$ we have: $g\circ f = g$, since
there is only one morphism from $0$ to $C$. Hence, $f$ is a right-identity.
Let $0$ and $0'$ be initial objects. Denote their right-identities as $f$ and $f'$ respectively. Let $g : 0 \rightarrow 0'$ and $g' : 0' \rightarrow 0$ be morphisms. Then $g'\circ g= f$ and $g \circ g' = f'$, that is $g$ is iso and $0 \cong 0'$.
The result for terminal objects follows from duality. Observe, that dually the endomorphism on a terminal object is a left-identity. $\square$

Is this correct and does it make sense? 
