Of course you can define a (just-arrow) category $\mathcal C$ like a partial algebra which consist of:

a set $\mathcal C$ (namely the set of arrows of your category), a set $D_\mathcal{C} \subseteq \mathcal C \times \mathcal C$ (the set of pair of composable arrows) and
a map $\circ \colon D_\mathcal{C} \to \mathcal C$, which is the composition for this "category".
In this structure we call identities all the elements $f \in \mathcal C$ such that for each $g,h \in \mathcal C$ with $(g,f),(f,h) \in D_\mathcal{C}$ we have $g\circ f=g$ and $f \circ h=h$.
The composition have to satisfy the following axioms:

*for each triple $h,g,f \in \mathcal C$ we have that these three statements are equivalent:

$(g,f) \in D_\mathcal{C}$ and $(h,g\circ f) \in D_\mathcal{C}$

$(h,g) \in D_\mathcal{C}$ and $(h\circ g, f) \in D_\mathcal{C}$

$(h,g) \in D_\mathcal{C}$ and $(g,f) \in D_\mathcal{C}$

and in this case the equality $h\circ(g \circ f)=(h \circ g) \circ f$ holds;

*for each $f \in \mathcal C$ there are two arrows $g,h \in \mathcal C$ which are identities such that $(f,g), (h,f) \in D$ and $f \circ g=f=h \circ f$.

With these data you have a concept of category just-arrow.
With this definition of category a functor $F$ from the category $\mathcal C$ to the category $\mathcal D$ is just a function $F \colon \mathcal C \to \mathcal D$ between the sets of the arrows such that:

for each pair $f,g \in \mathcal C$ if $(g,f) \in D_\mathcal{C}$ then $(\mathcal F(g),\mathcal F(f)) \in D_\mathcal{D}$ and $\mathcal F(g \circ f)= \mathcal F(g) \circ \mathcal F(f)$;

for each identity $f \in \mathcal C$ also $\mathcal F(f)$ is an identity.

The category of just-arrow categories and functors between them is proven to be equivalent to $\mathbf{Cat}$, the category of (ordinary) categories and functors between them.