Resolution of degenerate mirror families via toric morphisms

This paper continues the study of two examples of extremal transitions between families of Calabi-Yau threefolds. In a previous paper we suggested that the "mirror transition" between mirror families predicted by Morrison could be achieved naturally by combining a toric morphism with the Batyrev-Borisov construction. This was carried out for a particular example of a conifold transition. In this paper we show that similar methods work for another extremal transition involving more complicated singularities. We also study how the resolution is related to geometry of the ambient toric varieties, and discuss the connection with recent work by Doran and Harder.