I was reading many books about fiber bundles and they don't agree in a general definition of a fiber bundle map. The categories I was trying to understand were "vector bundles", "principal bundles" and "fiber bundles with structure group". Every fiber bundle with fiber $V$ vector space, is a vector bundle iff admits $GL(V)$ as structure group. I want a general notion of morphism that in this particular case reduces to vector bundle homomorphism.
An homomorphism of vector bundles are two smooth maps $F:E_1 \longrightarrow E_2$, $f:M_1 \longrightarrow M_2$, that $\pi_2 \circ F=f\circ \pi_1$ and $F|_{E_p}$ is linear $\forall p \in M_1$. For example if $f:M_1 \longrightarrow M_2$, $(df,f)$ is an homomorphism of $TM_1$ and $TM_2$.
A principal bundle map is a map $F_P:P_1 \longrightarrow P_2$ and a Lie group homomorphism $F_G:G_1 \longrightarrow G_2$, that $F_P(p\cdot g)=F_P(p)\cdot F_G(g)$. There is an unique smooth map in the base $F_M:M_1 \longrightarrow M_2$ that $\pi_2 \circ F_P=F_M \circ \pi_1$.
In bundle charts a principal bundle maps is of the form: $\psi\circ F_P \circ \xi^{-1}(m,g)=(F_M(m), \lambda_{\psi,\xi}(m)\cdot F_G(g))$, for some smooth $G_2$-valued function $\lambda_{\psi,\xi}(m)=pr_2(\psi\circ F_P \circ \xi^{-1}(m,1))$.
In analogy, I thought on a possible morphism between fiber bundles with structure group: $(G_1,F_1,E_1,\pi_1,M_1)$, $(G_2,F_2,E_2,\pi_2,M_2)$, an homomorphism are four maps, a Lie group homomorphism $F_G: G_1 \longrightarrow G_2$; an equivariant map $F_F: F_1 \longrightarrow F_2$, $F_F(g\cdot x)=F_G(g)\cdot F_F(x)$; $F_E: E_1 \longrightarrow E_2$ and $F_M: M_1 \longrightarrow M_2$ that $\pi_2 \circ F_E=F_M \circ \pi_1$. And the requirement that in charts, $\psi\circ F_E \circ \xi^{-1}(m,x)=(F_M(m), \lambda_{\psi,\xi}(m) \cdot F_F(x))$ for some $G_2$-valued function $\lambda_{\psi,\xi}$.
Lasts morphisms are stronger for vector bundles: they have constant rank. In particular not every map of the form $(f,df)$ is a morphism in the last sense.
In Poor's book, he defines a notion of subbundle involving a bundle chart condition. Stronger than requiring just $\pi_2=\pi_1|_{E_2}$ (with $\pi_1:E_1 \longrightarrow M_1$, $\pi_2:E_2 \longrightarrow M_2$, $E_2\subset E_1$, $M_2 \subset M_1$ submanifolds)