Summary: A Directed Acyclic Word Graph (DAWG) represents all factors of a string \(t\) over \(\Sigma\). By some isomorphism \(h: \Sigma^*\to (\sigma^q)^*\), with parameter \(q\), the string \(t\) can be transformed into \(h(t)\), the factors of which are represented by another DAWG, called the morphic DAWG, over a different alphabet \(\sigma\). Depending on \(h\), \(t\) has many morphic DAWGs, which exhibit time-space trade-off compared with the original DAWG. The bounds for storage reduction and time-space efficiency between DAWG and its morphic DAWGs implemented as tables are derived, as well as for their optimal points. Likewise, the storage increase, comparing the morphic DAWG implemented as tables and DAWGs implemented using bucket arrays, are also derived. The bounds of storage increase have global minimum when \(|\sigma|= 3\). Experiments show that round-off errors degrade storage reduction and time-space efficiency, so that \(q\) should be well chosen (i.e. \(|\sigma|^q= |\Sigma|\)). The compact morphic DAWG was found to be more space efficient than both the corresponding morphic DAWG and the compact DAWG. The compact morphic DAWGs over the binary alphabet require the least storage but the access speed is slower than those over the ternary alphabet. The space demand of compact morphic DAWGs is of similar order (within a factor of 3.3) of the bucket-array implementation of DAWGs.