  For a set-endofunctor $F$, we extend the notion of universal $F$-coalgebras
to $F$-graphs. These generalized coalgebras are models for various types of
graphs, such as (un)directed (hyper)graphs, relational structures or fuzzy
graphs. The induced morphisms coincide with graph homomorphisms. From this
point of view, graphs are "co-like" structures and share features of universal
coalgebras. In this article, we explore the coalgebraic character of graphs and
transfer coalgebraic concepts like cofreeness, simulations or Co-Birkhoff
theorems to $F$-graphs. Products and cofree constructions for $F$-graphs turn
out to be less restrictive than their coalgebraic counterparts.
