Am I correct that morphisms in a category are not necessary functional relations? That is, there can exist three objects $a$, $b$ and $c$ and an arrow $f$ such that we have both
$$a \xrightarrow{f} b \qquad\text{and}\qquad a \xrightarrow{f} c \quad ?$$
Do we still call arrows in such cases morphisms?
How do we call categories in which arrows are single-valued (or their opposite)?