To a ternary structure \((A,r)\) there corresponds a groupoid \((P(A),R[r])\) such that for \(X,Y\subseteq A\), \(R[r](X,Y)=\{z\in A:(\exists x\in X)(\exists y\in Y)(x,y,z)\in r\}\). Next, a binary hyperoperation \(o[r]\) on \(A\) is defined by putting \(o[r](x,y)=\{z\in A:(x,y,z)\in r\}\) for \(x,y\in A\). A pair \((A,o)\) is said to be a multiplicatively described ternary structure (MDT) if \(o=o[r]\) for some ternary relation \(r\) on \(A\). If \((A,o)\) is a MDT, then a binary multioperation \(O[o]\) is defined: if \(X,Y\subseteq A\), then \(O[o](X,Y)=\bigcup\{o(x,y): x\in X, y\in Y\}\). It is easy to see that \(O[o[r]]=R[r]\) for a tenary structure \((A,r)\) and \(O[o]=R[r[o]]\) for \((A,o)\) being a MDT. Let \((A,o)\), \((A^{'},o^{'})\) be MDTs, \(h\) be a mapping of \(A\) into \(A^{'}\). For \(X\subseteq A\) put \(h(X)=\{h(x):x\in X\}\). Then \(h\) is called a homomorphism (strong homomorphism) of \((A,o)\) into \((A^{'},o^{'})\) if for each \(X,Y\subseteq A\) the condition \(h(O[o](X,Y))\subseteq O[o^{'}](h(X), h(Y))\) (the condition \(h(O[o](X,Y)))=O[o^{'}](h(X),h(Y))\)) is satisfied. The main result of the paper is as follows: Let \((A,o)\), \((A^{'},o^{'})\) be MDTs. Then \(h:A\to A^{'}\) is a homomorphism (a strong homomorphism) of \((A,o)\) into \((A^{'},o^{'})\) iff \(h\) is a homomorphism (a strong homomorphism) of \((A,r[o])\) into \((A^{'},r[o])\). This implies that the category of all MDTs with homomorphisms (with strong homomorphisms) is isomorphic to the category of all ternary structures with homomorphisms (with strong homomorphisms). The notions and the results are applied to some classes of hypergroupoids.