I would like to know, in the context of exact sequences or short exact sequence:
$\ldots\stackrel{\rm }{\longrightarrow}H\stackrel{\rm g_{i-1}}{\longrightarrow}G\stackrel{\rm g_{i}}{\longrightarrow}K\stackrel{\rm }{\longrightarrow}\dots$
where $\text{Im }g_{i-1} = \text{Ker }g_{i}$. the term "morphism" as in "R-morphism" for modules, or group homo-morphisms for groups is always used instead of simply "maps" between algebraic objects of the same types for the sequence of arrows. Do the term "morphism" always mean homomorphism. I see it in math subjects where exact sequences make an appearance.
Thank you in advance
One uses the term `morphism' to mean a map which is defined between two objects of a given category. In the context you are talking about (homological algebra), one is usually working in an abelian category.
Morphisms in the category of groups are called group homomorphisms. In the category of $R$-modules, morphisms are called $R$-linear maps. In the category of topological spaces, the morphisms are called continuous maps. In the category of sets, the morphisms are the set maps.