In the following [K] refers to the paper http://dml.cz/bitstream/handle/10338.dmlcz/120545/ActaOstrav_07-1999-1_3.pdf.

That a split exact complex of projectives $(P,d)$ is a projective object can be seen as follows:

1. $im(d_n)$ is projective since it is a direct summand of $P_{n-1}$

2. By When is an acylic chain complex contractible a split exact complex is contractible, so $P$ is contractible.

3. By [K], Lemma 4.4 a contractible complex (like $P$) is isomorphic to the mapping cone of the boundary subcomplex $$ \cdots \to im(d_{n+1}) \xrightarrow{0} im(d_n) \to \cdots$$

4. By [K], Theorem 3.1, the mapping cone of a complex of projectives with zero differentials is a projective object. Hence $P$ is a projective object by 1. and 3.

In the following [K] refers to the paper https://dml.cz/bitstream/handle/10338.dmlcz/120545/ActaOstrav_07-1999-1_3.pdf (Kehrein, Achim Projective objects in the category of chain complexes. (English). Acta Mathematica et Informatica Universitatis Ostraviensis, vol. 7 (1999), issue 1, pp. 33-38).

That a split exact complex of projectives $(P,d)$ is a projective object can be seen as follows:

1. $im(d_n)$ is projective since it is a direct summand of $P_{n-1}$

2. By When is an acylic chain complex contractible a split exact complex is contractible, so $P$ is contractible.

3. By [K], Lemma 4.4 a contractible complex (like $P$) is isomorphic to the mapping cone of the boundary subcomplex $$ \cdots \to im(d_{n+1}) \xrightarrow{0} im(d_n) \to \cdots$$

4. By [K], Theorem 3.1, the mapping cone of a complex of projectives with zero differentials is a projective object. Hence $P$ is a projective object by 1. and 3.

In the following [K] refers to the paper http://dml.cz/bitstream/handle/10338.dmlcz/120545/ActaOstrav_07-1999-1_3.pdf.

That a split exact complex of projectives $(P,d)$ is a projective object can be seen as follows:

1. $im(d_n)$ is projective since it is a direct summand of $P_{n-1}$

2. By When is an acylic chain complex contractible a split exact complex is contractible, so $P$ is contractible.

3. By [K], Lemma 4.4 a contractible complex (like $P$) is isomorphic to the mapping cone of the boundary subcomplex $$ \cdots \to im(d_{n+1}) \xrightarrow{0} im(d_n) \to \cdots$$

4. By [K], Theorem 3.1, the mapping cone of a complex of projectives with zero differentials is a projective object. Hence $P$ is a projective object by 1. and 3.

In the following [K] refers to the paper http://dml.cz/bitstream/handle/10338.dmlcz/120545/ActaOstrav_07-1999-1_3.pdf.

That a split exact complex of projectives $(P,d)$ is a projective object can be seen as follows:

1. $im(d_n)$ is projective since it is a direct summand of $P_{n-1}$

2. By When is an acylic chain complex contractible a split exact complex is contractible, so $P$ is contractible.

3. By [K], Lemma 4.4 a contractible complex (like $P$) is isomorphic to the mapping cone of the boundary subcomplex $$ \cdots \to im(d_{n+1}) \xrightarrow{0} im(d_n) \to \cdots$$

4. By [K], Theorem 3.1, the mapping cone of a complex of projectives with zero differentials is a projective object. Hence $P$ is a projective object by 1. and 3.