Connecting pins of L293D to pins of inverter 7404
If you really must avoid the use of the inverter IC, an alternative is to create your own inversion in software, using more of the Arduino pins: 

Each of the outputs generated by your Arduino code is being used to generate a positive and a matching negative signal via the inverter, to drive opposing halves of the H-Bridge in the L293.

To achieve the same result in software, if your code uses something like DigitalWrite(pin1, x) on a given pin, you will have to assign another pin and add another  DigitalWrite(pin2, !x) immediately following the previous line of code. This will generate the inverted signal, which you would feed in where the inverter output line goes in your current circuit.

If, however, PWM and AnalogWrite() are used as your method of motor control, some slightly more involved coding will be needed to generate PWM that is exactly opposite in phase to the existing PWM outputs. This may well be within your skills (or not), but I'll refrain from coding it for you in this answer. 

If you need help with that, it deserves a separate question.