Let \(X\), \(G\), \(Y\), \(H\) be Hausdorff topological spaces and \(S(X,G,\alpha)\), \(S(Y,H,\beta)\) be sandwich semigroups of continuous functions with continuous sandwich functions \(\alpha: G \to X\), \(\beta : H \to Y\), respectively. If \(\varphi: S(X,G,\alpha) \to S(Y,H,\beta)\) is a homomorphism and for suitable mappings \(h: Y \to X\), \(t : G \to H\) we have \(\alpha = h\circ \beta \circ t\), \(\varphi(f) = h \circ f\circ t\) for any \(f\in S(X,G,\alpha)\), then \(\varphi\) is said to be induced by the functions \(h\) and \(t\). There are characterized monomorphisms of \(S(X,G,\alpha)\) into \(S(Y,H,\beta)\) under some special assumptions concerning the space \(X\) and also other special morphisms between the sandwich semigroups in question, as well as, between sandwich near-rings \(N(X,G,\alpha)\), \(N(Y,H,\beta)\) in the case when \(G\), \(H\) are additive topological groups.