Assalamualaikum ustaz. I arrive early at the mosque and perform the congregational prayer with the imam. Then, when I have yet completed by Fatihah recitation, the imam has already bowed. Should I finish and complete my Fatihah recitation?
Waalaikumussalam wrt. wbt.,
Alhamdulillah, praise and gratitude to Allah SWT for His countless blessings for us all. Praise and salutations to our beloved Prophet Muhammad PBUH, his family, companions, and all those who follow his footsteps until the Final day.
Prayer has its own requisites. It is divided into fi’li requisites which are related to actions, qauli requisites which is related to speech and recitation, qalbi requisites and maknawi requisites. All of these requisites are obligatory to be performed for the prayer to be valid.
Reciting al-Fatihah is one of the qauli requisites. A prayer is invalid without the recitation of al-Fatihah. This follows the statement of the Prophet PBUH:
لاَ صَلاَةَ لِمَنْ لَمْ يَقْرَأْ فِيهَا بِفَاتِحَةِ الْكِتَابِ
“The prayer is not valid I one does not recite Fatihat al-kitab,” 
In explaining this hadith, Imam al-Nawawi: “Reciting al-Fatihah for those who are capable to recite it is an obligation and requisite of which it cannot be translated into any language other than Arabic and it can’t be replaced with any other recitation from al-Quran. It is also obligatory to be recited in all prayers whether it is for obligatory or sunnah prayer.” 
This means that prayer is invalid if al-Fatihah isn’t recited in prayer. If a person forgets to recite al-Fatihah, then the obligation is abrogated from him. This opinion is agreed and concluded by jumhur of scholars in madhhab Syafie. This is stated by Imam al-Nawawi. 
Coming back to the above question as explained in madhhab Imam al-Syafie it is obligatory for the makmum to recite al-Fatihah in prayer. This is based on a general hadith where the Prophet PBUH said:
لاَ صَلاَةَ لِمَنْ لَمْ يَقْرَأْ بِفَاتِحَةِ الْكِتَابِ
“He who does not recite Fatihah al-Kitab is not credited with having observed prayer.” 
Imam al-Nawawi said:
مَذْهَبُنَا وُجُوبُ قِرَاءَةِ الْفَاتِحَةِ عَلَى المَأْمُومِ فِي كُلِّ الرَّكعَاتِ مِنَ الصَّلَاة السِّرِّيَّةِ وَالجَهْرِيَّةِ وَهَذَا هُو الصَّحِيحُ عِنْدَنَا كَمَا سَبَق وَبِهِ قَالَ أَكْثَرُ العُلَمَاء
“In our madhhab, in our opinion, the makmum is obligated to recite al-Fatihah in every rakaat whether it is al-sirriyah prayer (a prayer where the recitation is recited in a low voice) or al-jahriyyah prayer (its recitation is recited loudly) and this is the sahih opinion that has been previously stated and this is also the opinion of the majority of scholars.” 
In this case, in our opinion, the makmum has to finish his al-Fatihah recitation with a moderate pace of not too fast or slow.
One thing that should be paid attention to for the imam is to celebrate the situation of the makmum behind him when praying not to be too fast that the makmum wouldn’t be able to finish their al-Fatihah recitation.