Groupoids, i.e., algebras \((A,\oplus )\) with one binary operation \(\oplus\), are considered under the name “magmas”. If \(\oplus\) is commutative, cancellative and medial (the last one means \((x\oplus y)\oplus (z\oplus w)=(x\oplus z)\oplus (y\oplus w)\)) the magma \((A,\oplus )\) is called a ccm-magma. Several examples of ccm-magmas are presented. The authors study the category of ccm-magmas. This category is weakly Mal’tsev. The existence of an internal monoid structure \(A\times A \to A \leftarrow 1\) is characterized for a given object \((A,\oplus )\) and for every choice of a unit element. Some properties of internal relations are studied.